Although the other languages had local rather than global prominence.
True, but those localities were rather large. Aramaic covered all of "Western" civilization at first, and continued to be the
lingua franca of western Asia through the Greek, Roman, Arab and Ottoman hegemonies. When the Middle East broke off from Western civilization, Latin was the language of government, scholarship and religion for Europe until quite recently.
Yes there were several other civilizations but despite their impressive cultures which were in some ways arguably superior, they lagged behind Europe in the key areas of scientific and economic progress. The Greco-Roman model and the ever-growing region it defines came to dominate world civilization. So despite the billions of people who speak Chinese, Hindi and Arabic, English for better or worse is the world's most important language, like Latin and Greek before it.
The dominance of English was backed by two major technological shifts: first the Industrial Revolution which spread from Britain, and then the Internet which spread from the USA.
Which is an offshoot of Britain. We regard King Arthur, Shakespeare, Robin Hood and the Beatles as "our culture" and until the end of time Americans will always die to protect England.
For how long have we been able to send messages around the world at effectively zero cost? It's difficult to imagine the world before then.
Not for us older people.

Most of my Esperanto "pen pals" in the former Soviet bloc still haven't got e-mail.
Do you think there will be a new revolution favoring another language, or will it just be a gradual power shift this time?
One of the things you learn from history is that it's difficult to make predictions across a paradigm shift, such as the transition from the Industrial Era into the Information Age. English is rapidly becoming the world language that Esperanto aspired to be. Despite being economic powerhouses, the Chinese and Japanese people are not promoting their languages for the obvious reason that they're horrible for a computer keyboard. India is fast becoming an economic powerhouse, but linguistically it's an English colony. Latin America is proud of its languages, but it will never discard Spanish for Portuguese or vice versa, so it can't present a united front.
It's quite possible that English will continue to spread because it has so much inertia behind it. How many billions of lines of computer code--civilization's new infrastructure--are documented only in English? By the time America's hegemony fades away, its language may be the only one in which business and science can be conducted. Very much like Latin after the fall of Rome.
Or perhaps Aramaic is a better analogy because of its strange history. The Aramaeans were just one of the peoples the Assyrians conquered, but their language began to spread among the neighboring captive tribes. Eventually the Assyrians made it the official language of the empire. After their empire waned, Aramaic continued to be spoken throughout the Middle East for two thousand years, despite the comings and goings of various ethnic groups and the rise and fall of empires. Whether the people of the Middle East were ruled by Persians, Greeks, Tatars, Moghuls, Arabs, Ottomans or Britons, through sheer inertia they kept speaking Aramaic.